The linear invariants (DN) and $(\overline{\Omega})$ for spaces of germs of holomorphic functions on compact subsets of $\mathbf{C}^n$

Bui Dac Tac, Le Mau Hai, Nguyen Quang Dieu

DOI: 38088

Resum

For a compact subset $K$ of $\mathbf{C}^n$, we give necessary and sufficient conditions for $[\mathcal{H}(K)]'$ to have the property $(\underline{\mathit{DN}})$, and similarly for the property $(\overline{\Omega})$. We also show that $\mathcal{H}(\overline{D})$ is isomorphic to $\mathcal{H}(\overline{\Delta}^n)$, where $\Delta^n$ is the unit polydisc in $\mathbf{C}^n$ and $D$ is any bounded Reinhardt domain in $\mathbf{C}^n$. This last result requires a generalization of the classical Hartogs phenomenon.